Editing Representable functor (section)

==Examples==

* The [[functor represented by a scheme]] ''A'' can sometimes describe families of geometric objects''.'' For example, [[Vector bundle|vector bundles]] of rank ''k'' over a given algebraic variety or scheme ''X'' correspond to algebraic morphisms <math>X\to A</math> where ''A'' is the [[Grassmannian]] of ''k''-planes in a high-dimensional space. Also certain types of subschemes are represented by [[Hilbert scheme|Hilbert schemes]]. 
* Let ''C'' be the category of [[CW-complex]]es with morphisms given by homotopy classes of continuous functions. For each natural number ''n'' there is a contravariant functor ''H''<sup>''n''</sup> : ''C'' → '''Ab''' which assigns each CW-complex its ''n''<sup>th</sup> [[cohomology group]] (with integer coefficients). Composing this with the [[forgetful functor]] we have a contravariant functor from ''C'' to '''Set'''. [[Brown's representability theorem]] in algebraic topology says that this functor is represented by a CW-complex ''K''('''Z''',''n'') called an [[Eilenberg–MacLane space]].
*Consider the contravariant functor ''P'' : '''Set''' → '''Set''' which maps each set to its [[power set]] and each function to its [[inverse image]] map. To represent this functor we need a pair (''A'',''u'') where ''A'' is a set and ''u'' is a subset of ''A'', i.e. an element of ''P''(''A''), such that for all sets ''X'', the hom-set Hom(''X'',''A'') is isomorphic to ''P''(''X'') via Φ<sub>''X''</sub>(''f'') = (''Pf'')''u'' = ''f''<sup>−1</sup>(''u''). Take ''A'' = {0,1} and ''u'' = {1}. Given a subset ''S'' ⊆ ''X'' the corresponding function from ''X'' to ''A'' is the [[indicator function|characteristic function]] of ''S''.
*[[Forgetful functor]]s to '''Set''' are very often representable. In particular, a forgetful functor is represented by (''A'', ''u'') whenever ''A'' is a [[free object]] over a [[singleton set]] with generator ''u''.
** The forgetful functor '''Grp''' → '''Set''' on the [[category of groups]] is represented by ('''Z''', 1).
** The forgetful functor '''Ring''' → '''Set''' on the [[category of rings]] is represented by ('''Z'''[''x''], ''x''), the [[polynomial ring]] in one [[Variable (mathematics)|variable]] with [[integer]] [[coefficient]]s.
** The forgetful functor '''Vect''' → '''Set''' on the [[category of real vector spaces]] is represented by ('''R''', 1).
** The forgetful functor '''Top''' → '''Set''' on the [[category of topological spaces]] is represented by any singleton topological space with its unique element.
*A [[group (mathematics)|group]] ''G'' can be considered a category (even a [[groupoid]]) with one object which we denote by •. A functor from ''G'' to '''Set''' then corresponds to a [[G-set|''G''-set]]. The unique hom-functor Hom(•,&ndash;) from ''G'' to '''Set''' corresponds to the canonical ''G''-set ''G'' with the action of left multiplication. Standard arguments from group theory show that a functor from ''G'' to '''Set''' is representable if and only if the corresponding ''G''-set is simply transitive (i.e. a [[torsor|''G''-torsor]] or [[heap (mathematics)|heap]]). Choosing a representation amounts to choosing an identity for the heap.
*Let ''R'' be a commutative ring with identity, and let '''R'''-'''Mod''' be the category of ''R''-modules. If ''M'' and ''N'' are unitary modules over ''R'', there is a covariant functor ''B'': '''R'''-'''Mod''' → '''Set''' which assigns to each ''R''-module ''P'' the set of ''R''-bilinear maps ''M'' × ''N'' → ''P'' and to each ''R''-module homomorphism ''f'' : ''P'' → ''Q'' the function ''B''(''f'') : ''B''(''P'') → ''B''(''Q'') which sends each bilinear map ''g'' : ''M'' × ''N'' → ''P'' to the bilinear map ''f''∘''g'' : ''M'' × ''N''→''Q''. The functor ''B'' is represented by the ''R''-module ''M'' ⊗<sub>''R''</sub> ''N''.<ref>{{cite book|last1=Hungerford|first1=Thomas|title=Algebra|publisher=Springer-Verlag|isbn=3-540-90518-9|page=470}}</ref>